Characterizations of a family of bivariate Pareto distributions

CHARACTERIZATIONS OF A FAMILY OF BIVARIATE

PARETO DISTRIBUTIONS

P. G. Sankaran1

Department of Statistics Cochin University of Science and Technology, Cochin N. Unnikrishnan Nair

Department of Statistics Cochin University of Science and Technology, Cochin Preethi John

Department of Statistics Cochin University of Science and Technology, Cochin

1. Introduction

In the literature, Pareto distributions have been extensively employed for modeling and analysis of statistical data under different contexts. Orginally, the distribution was first proposed as a model to explain the allocation of wealth among individuals. Later, various forms of the Pareto distribution have been formulated for modeling and analysis of data from engineering, environment, geology, hydrology etc. These diverse applications of the Pareto distributions lead researchers to develop different kinds of bivariate(multivariate)Pareto distributions. Accordingly, Mardia (1962) introduced two types of bivariate(multivariate) Pareto models which are referred as bivariate Pareto distributions of first kind and second kind respectively. Since then there has been a lot of works in the form, alternative derivation of bivariate Pareto models, their extensions, inference, characterizations and applications to a variety of fields. Various types of bivariate (multivariate) Pareto distributions discussed and studied in literature include those of Lindley and Singpurwalla (1986), Arnold (1985), Arnold (1990), Sankaran and Nair (1993), Hutchinson and Lai (1990), Langseth (2002), Balakrishnan and Lai (2009) and Sankaran and Kundu (2014).

The models discussed above are individual in nature and are appropriate for a particular data set that meet the specified requirements. However, when there is little information about the data generating process, it is desirable to start with a family of distributions and then choose a member of the family that agrees with the patterns in the data. Motivated by this fact, we introduced a family of bivariate Pareto distributions arising from a generalization of the univariate dull-ness property (Talwalker (1980)) that characterized the univariate Pareto law.(see Sankaranet al. (2014))

The major aim of the present paper is to develop characterizations of the above family of Pareto distributions. One of the problems that is usually addressed while examining the characteristic properties of the bivariate distribution is to investi-gate how far the characterizations of the corresponding univariate version can be extended to the bivariate forms. Among the characterizations of the univariate Pareto I law, one with important practical applications is the dullness property. The bivariate version of the dullness property is employed to characterize the family of Pareto distributions.

A second concept that has applications in economics is income gap ratio which is used for developing indices of affluence (Sen (1988)). The bivariate general-ization of the concept is proposed and charactergeneral-izations using this concept are derived. Another function of interest that has applications in reliability also is the bivariate generalized failure rate. Characterizations of the family of distributions using the generalized failure rate are also discussed. As the bivariate versions of these functions are not unique, different versions of these concepts lead to various types of bivariate Pareto distributions which are members of the family.

The rest of the paper is organized as follows. In Section 2, we present a family of bivariate Pareto distributions characterized through a generalized version of the univariate dullness property. In Section 3, we introduce bivariate versions of dullness property and present characterizations using these versions. The bivariate version of income gap ratio and related concepts in economics are discussed in Section 4. The forms of these functions for various bivariate Pareto distributions are given. Section 5 discusses bivariate generalized failure rate. Characterizations using the generalized failure rate are also developed. Finally, Section 6 provides brief conclusions of the study.

2. Bivariate Pareto family

Let (X, Y) be a non-negative random vector having absolutely continuous survival function ¯F(x, y) = P(X > x, Y > y). Assume thatZ is a non-negative random variable with continuous and strictly decreasing survival function ¯G(z) and cumu-lative hazard function H(z) defined by H(z) = −log ¯G(z). Then the family of distributions specified by the survival function

¯

F(x, y) = [g(x, y)]−1, x, y >1 (1)

is a bivariate Pareto family if and only if there exist a functiong(x, y) satisfying

H(logg(x, y)) = H(alogx) +H(blogy) (2)

Notice thatg(x, y) is a function of (x, y) inR+₂ ={(x, y)|x, y >0} satisfying the following properties

(a) g(1, y) =yb_{, g(x,1) =x}a_,

(b) g(∞, y) =∞, g(x,∞) =∞,

(d) it is assumed thatg(x, y) satisfies the inequality 2 g(x,y)

∂g ∂x

∂g ∂y −

∂2g ∂x∂y ≥0.

Further, (2) holds if and only if

P(Z >logg(x, y)|Z > alogx) =P(Z > blogy). (3)

The proof of this is given in Sankaranet al.(2014). Notice that the marginal dis-tributions ofX andY are Pareto I distributions with survival functions ¯FX(x) =

x−a_{, x > 1 and ¯F}

Y(y) = y−b, y > 1. The property (3) is referred as extended

dullness property. The family (1) includes various known and unknown bivariate distributions. The members of the family can be derived by choosing appropriate univariate distribution ofZ. Table 1 provides some members of family according to different choices of the distribution of Z. It may be noted that the proper-ties of ¯F(x, y) can be inferred from the corresponding properties of ¯G(z). The members of the family listed in Table 1, include bivariate Pareto with indepen-dent marginals, Mardia’s(1962) Type 1 model and bivariate Burr distribution. For more properties one could refer to Sankaranet al.(2014).

3. Bivariate dullness property

We first discuss the univariate dullness property. LetZ1be a non-negative random variable representing the income in a population. Then the distribution ofZ1 is said to have dullness property if

P(Z1> xy|Z1> x) =P(Z1> y) (4)

for allx, y≥1.This means that the conditional probability that true incomeZ1is at leasty times the reported valuexis the same as the unconditional probability thatZ1has at least incomey. In otherwords, the distribution of error in an income is independent of the reported value. It is proved that the property (4) holds if and only if the distribution ofZ1is Pareto I (Talwalker (1980)). There have been many works on models that accounts for underreporting, characterizations and related concepts, like those of Wong (1982), Xekalaki (1983), Korwar (1985), Artikiset al. (1994), Pham-Gia and Turkkan (1997) and Nadarajah (2009). Sometimes it is more convenient to work with an equivalent form of dullness property, as the property (4) is not helpful in practice to verify whether a given data set follows Pareto distribution or not. In this context, we have the following result.

Theorem 1. The random variable Z1 satisfies dullness property (4) if and only if

m(x) =E(Z1|Z1> x) =µx (5)

for allx >1, whereµ=E(Z1)<∞.

Proof. Assume that (4) holds. Then we can write

278

P.

G.

Sankar

an,

N.

Unnikrishnan

Nair

and

Pr

eethi

TABLE 1

Some members of bivariate Pareto family

Type Survival function of the distribution ofZ Survival function of the bivariate Pareto distribution 1 G¯1(z) = exp(−λ z), z >0 F¯1(x, y) =x−ay−b; x > y >1;a, b >0

2 G¯2(z) = exp[−θ(eαz−1)];z≥0;α, θ >0 F¯2(x, y) = (xaα+ybα−1) −1

α ;x, y >1, α, a >0

3 G¯4(z) = (1 +βz)−α F¯4(x, y) =x−a−clogyy−b, x, y >1, a, b >0; 0≤c≤1

4 G¯5(z) = 2(1 +e

z

σ)−1, z >0, σ >0 F¯₅(x, y) = [1

2(x

α_+yβ_+xα_yβ_−1)]−σ_;α= a

σ >0, σ >0, β= b σ >0

5 G¯6(z) = (1 +zc)−k, z >0; c, k >0 F¯6(x, y) = exp[−(alogx)c−(blogy)c−(ablogxlogy)c] 1

c

6 G¯7(z) = (2ez−1)−σ, z >0;σ >0 F¯7(x, y) = (1 + 2xayb−xa−yb)−1

7 G¯8(z) =e−(λz)

α

α, λ >0, z >0 F¯8(x, y) = exp[−_λ1{(λalogx)α+ (λblogy)α} 1

α]

8 G¯9(z) = _eλzp_−q, z >0;λ >0, 0< p <1, q= 1−p F¯9(x, y) = (q+p−1(xaλ−q)(ybλ−q)) −1

λ

9 G¯10(z) = (1 +e

λz₋₁

α )−

1_{, α, λ >0 F¯}

10(x, y) = (1 +α−1(α+xaλ−1)(α+ybλ−1)−α) −1

whereS(x) =P(Z1> x). Integrating (6) from 1 to∞, we obtain

∞ ∫

1

S(xy)dy= S(x)

∞ ∫

1

S(y)dy

or

∞ ∫

x

S(t)dt= x S(x)(µ−1) (7)

Since left side of (7) is E(Z1−x|Z1> x), we get

m(x)−x=x(µ−1)

which leads to (5).

Conversely, when (5) hold, we obtain (7) and hence

S(x)

∞ ∫

x

S(t)dt

= 1

x(µ−1) (8)

Integrating (8), we obtain

∞ ∫

x

S(t)dt=cx−µ−11 ₍₉₎

wherec is the integrating constant. Differentiating (9) with respect tox, we get

S(x) = c

µ−1x

− µ

µ−1 (10)

SinceS(1) =1, we obtainc=µ−1 and thus

S(x) =x−µµ−1

which means that the distribution of Z1 is Pareto I. From Talwalker (1980) it follows that the Pareto distribution satisfies (4), which completes the proof. 2

Now we propose bivariate versions of (5) and examine whether they charac-terize members of the bivariate Pareto family. A natural extension of (5) with respect to the random vector (X, Y) is the vector (m1(x, y), m2(x, y)) where

m1(x, y) =E(X|X > x, Y > y) (11)

and

The joint survival function ¯F(x, y) is obtained as

¯

F(x, y) = exp

 ₋

x

∫

1

∂m1(t,1) ∂t

m1(t,1)−t

dt−

y

∫

1

∂m2(x,t) ∂t

m2(x, t)−t

dt 

 (13)

= exp

 ₋

y

∫

1

∂m2(1,t) ∂t

m2(1, t)−t

dt−

x

∫

1

∂m1(t,y) ∂t

m2(t, y)−t

dt 

 (14)

The proof follows from Nair and Nair (1989) by using the relationship between (m1(x1, x2), m2(x1, x2)) and bivariate mean residual life functions.

Definition 2. The distribution of the random vector(X, Y) is said to have bivariate dullness property one (BDP-1) if and only if forx, y >1

P(X > xt|X > x, Y > y) =P(X > t|Y > y), t >1

and

P(Y > ys|X > x, Y > y) =P(Y > s|X > x), s >1.

Definition 3. The distribution of(X, Y)satisfies bivariate dullness property two (BDP-2) if and only if for x, y >1

P(X > xt, Y > ys|X > x, Y > y) =P(X > t, Y > s). (15)

Now we have the following theorems characterizing bivariate Pareto distributions belonging to the family by two versions of the bivariate dullness property.

Theorem 4. Let (X, Y)be a bivariate random vector as described in Section 2, with µX =E(X)<∞ and µY =E(Y)<∞. Denote µX(y) =E(X|Y > y) andµY(x) =E(Y|X > x). Then the following statements are equivalent.

(a) F¯4(x, y) =x−(a+clogy)y−b; x, y >1, a, b >1, 0< c≤a b (b) (X, Y)satisfies BDP-1.

(c) m1(x, y) =xµX(y) andm2(x, y) =yµY(x)

Proof. To prove (a)⇒(b), we have

P[X > xt|X > x, Y > y] = (xt)

−(a+clogy)_y−b

x−(a+clogy)_y−b =t

−(a+clogy)_{=P(X > t|Y > y)}

The second part is proved similarly.

To establish (b)⇒(c), we note that the first statement in (b) is same as ¯

F(xt, y) ¯

F(x, y) = ¯

F(t, y)

P(Y > y) = ¯FX(t|Y > y) (16)

where ¯FX(t|Y > y) is the conditional survival function of X givenY > y.

Inte-grating (16) with respect totin (1,∞), we obtain

or

m1(x, y) =xµX(y).

The expression form2(x, y) is proved similarly. Now we establish (c)⇒(a). From (c), we have

∂m1(t,1)

∂t = ∂

∂t(tµX(y)) =µX, ∂m2(1, t)

∂t =µY

m1(t, y)−t=tµX(y)−t

and

m2(x, t)−t=tµY(x)−t.

Substituting the above expressions in (13) and (14) and equating the resulting expressions, we get, after some simplifications,

µX

µX−1

logx+ µY(x)

µY(x)−1

logy= µY

µY −1

logy+ µX(y)

µX(y)−1

logx

which leads to the functional equation

( µX

µX−1−

µX(y)

µX(y)−1

)

logx=

( µY

µY −1−

µY(x)

µY(x)−1

)

logy (17)

To solve (17), we rewrite it as

logx

µY

µY−1−

µY(x)

µY(x)−1

= logy

µX

µX−1−

µX(y)

µX(y)−1

(18)

The right(left) side of (18) is a function ofy(x) alone and therefore the equality of the two sides hold good for allx, y >1 if and only if each side must be a constant say 1_c. Hence

µY(x)

µY(x)−1

= µY

µY −1 −

clogx

and

µX(y)

µX(y)−1

= µX

µX−1 −

clogy (19)

Using (19) in (13) or (14), we get

¯

F4(x, y) =x− µX

µX−1_y−µYµY−1+clogx

TakingµX= _a−a₁ andµY = _b−b₁ ,a, b >1, we obtain

¯

F4(x, y) =x−ay−(b+clogx)

sincex−clogy_=y−clogx_{, we have (a) and the theorem is completely proved. The}

Settingc= 0 in ¯F4(x, y) and working similarly, we have the following result.

Theorem 5. For the random vector(X, Y)in Theorem 4, the following state-ments are equivalent

(a) F¯1(x, y) =x−ay−b; x, y >1, a, b >1 (b) (X, Y)satisfies BDP-2.

(c) (m1(x, y), m2(x, y)) = (xµX, yµY)

Remark 6. It may be noticed thatBDP −1 is stronger thanBDP −2. Remark 7. The propertiesBDP−1andBDP−2can be interpreted in income analysis as follows. Let X andY be the incomes from two different sources of a unit in a population. Assume that the incomes of X and Y are at least x and

y respectively. The average under-reporting error is proportional to the amount by which the income exceeds the tax exemption level. The under-reporting error in X(Y) is a linear function of the reported income if and only if the incomes (X, Y)follow bivariate Pareto law. In the case ofBDP−1, the proportionality is independent ofx andy, while in BDP −2, it is independent of xin the case of

X and independent of y in the case ofY.

Remark 8. Hanagal (1996) has considered another version of the dullness property defined as

P(X > xt, Y > yt) =P(X > x, Y > y)P(X > t, Y > t)

which characterized a Marshall-Olkin type bivariate Pareto model. Note that this distribution does not belongs to the family (1).

Theorem 9. Let (X, Y) be a non-negative exchangeable random vector with absolutely continuous survival function andµ=E(X) =E(Y)<∞. Then

(m1(x, y), m2(x, y)) = (µx+ (µ−1)p(y), µy+ (µ−1)p(x)) (20) for some non-negative function p(.) with p(1) = 0 iff the survival function of (X, Y)is

¯

F10(x, y) =

(

x+y−cxy−1 1−c

)₋ µ µ−1

; x, y >1 (21)

wherec is a real constant different from unity.

Proof. Assume that (X, Y) has the distribution (21). Then

m1(x, y) =x+ 1 ¯

F(x, y)

∞ ∫

x

¯

F(t, y)dt

=x+x+y−cxy−1

1−cy (µ−1)

=µx+ y−1

which is of the form (20) withp(y) = ₁y₋−_cy1 andp(1) = 0. The proof form2(x, y) is similar.

Conversely, if the relation (20) holds, from (14),

¯

F(x, y) = exp

 ₋

x

∫

1

µ

(µ−1)tdt−

y

∫

1

µ

(µ−1)t+ (µ−1)p(x)dt

 

=

(

x(y+p(x)) 1 +p(x)

)₋ µ µ−1

. (22)

From (14) and (20), we obtain

¯

F(x, y) =

(

y(x+p(y)) 1 +p(y)

)− µ µ−1

. (23)

Equating (22) and (23) and simplifying,

xp(x) 1 +p(x)−x=

yp(y)

1 +p(y)−y. (24)

Since (24) holds for allx, y >1, one should have

xp(x) 1 +p(x)−x =

1

c,

a constant independent of x and y. Solving the above, we get

p(x) = x−1 1−cx.

Substitutingp(x) in (22), we have (21). This completes the proof. 2

Remark 10. The distribution specified by (21)is a bivariate distribution with Pareto I marginals. It contains some members of our family. When c= 0

m1(x, y) =µx+ (µ−1)(y−1)

and

m2(x, y) =µy+ (µ−1)(x−1) characterized the bivariate Pareto distribution

¯

F3(x, y) = (x+y−1)

−a

;x, y >1,

the well known Mardia’s(1962) type I bivariate Pareto model. Similarly when

c=−1, we have

(m1(x, y), m2(x, y)) =

(

µx+ (µ−1)y−1

y+ 1, µy+ (µ−1)

x−1

x+ 1

characterized the bivariate Pareto distribution

¯

F11(x, y) = [ 1

2(x+y+xy−1)]

−a_{, x, y >1, a >1}

which is a special case of F¯5(x, y) in Table 1 when α = β = 1 so that σ = a. Finallyc= 1_q, q >0 gives

m1(x, y) =µx+

(µ−1)q(y−1)

q−y

and

m2(x, y) =µy+

(µ−1)q(x−1)

q−x

that characterizes

¯

F12(x, y) = (q+p−1(x−q)(y−q))−a

a special case ofF¯9(x, y)obtained by taking λa=λb= 1.

Remark 11. It is easy to see that all the bivariate distributions discussed in Remark 10, including our modelsF¯3(x, y),F¯5(x, y)andF¯9(x, y)do not satisfy the dullness properties BDP −1andBDP −2.

The extent to which they depart fromBDP−1is accounted for by the termsµp(x) andµp(y).

Remark 12. A closely related function to(m1(x, y), m2(x, y))used extensively in reliability analysis is the bivariate mean residual life function (see Nair and Nair, 1988) defined as

(E(X−x|X > x, Y > y), E(Y −y|X > x, Y > y)) = (m1(x, y)−x, m2(x, y)−y) (25) It follows that Theorems 4 through 9 provide useful characterizations of the con-cerned distributions by the form of the bivariate mean residual life function that can be easily deduced from the relationship (25).

4. Bivariate income gap ratio

In the context of applications in economics, in the univariate case, two functions that are closely related tom(x) are the income gap ratio and the left proportional residual income. For a continuous non-negative random variable Z which repre-sents the income of a population, those with income exceedingxare deemed to be affluent or rich. We call Z =xto be the affluence line. Then ¯G(z) = P(Z > z) represents the proportion of rich in the population. The proportion of rich, their average income and the measures of income inequality are important indices dis-cussed in connection with income analysis and also for comparison between the rich and poor. Of these, Sen (1988) defines the income gap ratio among the affluent as

i(x) = 1− x

TABLE 2

Bivariate income gap ratios

Distribution

(

i

(

x, y

)

, i

(

x, y

))

¯

F

(

x, y

)

(

µX−1

µX

,

µY

)

¯

F

(

x, y

)

(

µX(y)−1

µX(y)

,

µY(x)−1

µY(x)

)

¯

F

(

x, y

)

(

(µ−1)(x+y−2)

µx+(µ−1)(y−1)

,

(µ−1)(x+y−2)

µx+(µ−1)(y−1)

)

¯

F

(

x, y

)

(

(µ−1)(x+y+xy−1)

µx(y+1)+(µ−1)(y−1)

,

(µ−1)(x+y+xy−1)

µx(y+1)+(µ−1)(y−1)

)

¯

F

(

x, y

)

(

(µ−1)(x(q−y)+q(y−1))

µx(q−y)+q(µ−1)(y−1)

,

(µ−1)(y(q−x)+q(x−1))

µx(q−y)+q(µ−1)(y−1)

)

The measure i(x) is used in defining indices of affluence in Sen (1988). On the other hand, Belzunce et al. (1998) defined the mean left proportional residual income(M LP RI) as

l(x) =E (

X x|X > x

)

= 1− 1

i(x). (27) We propose bivariate generalization of these concepts. For a non-negative random vector (X, Y), the bivariate income gap ratio is defined by the vector

(i1(x, y), i2(x, y)) = (1−

x

E(X|X > x, Y > y),1−

y

E(Y|X > x, Y > y)) = (1− x

m1(x, y)

,1− y

m2(x, y)

). (28)

Equation (28) shows that there is one-to-one relationship between (i1(x, y), i2(x, y)) and (m1(x, y), m2(x, y)), so that each determine other and the corresponding dis-tribution uniquely. The functional forms of (i1(x, y), i2(x, y)) characterizing some members of our family are given in Table 2.

The bivariate generalization ofM LP RI is proposed as the vector

(l1(x, y), l2(x, y)) =

( E

( X

x|X > x, Y > y )

, E (

Y

y|X > x, Y > y ))

= (m1(x, y)

x ,

m2(x, y)

y ) (29)

The calculation of (l1(x, y), l2(x, y)) is easily facilitated from those of (m1(x, y), m2(x, y)). Thus the characterizations established in Section 3 using (m1(x, y), m2(x, y)) can be translated in terms of (l1(x, y), l2(x, y)).

5. Bivariate generalized failure rate

For a non-negative random variableZ, the generalized failure rate is given by

r(x) =−xdlog

¯

G(x)

Lariviere and Porteus (2001) and Lariviere (2006) discussed the properties ofr(x) and its applications in operations management. The well known income model derived by Singh and Maddala (1976) is based on a relationship betweenr(x) and

¯

G(x) as

r(x) =αxβ( ¯G(x))γ, α, β, γ >0.

For a non-negative random vector (X, Y), the bivariate generalized failure rate vector is defined by

(r1(x, y), r2(x, y)) =

(

−x∂log ¯F(x, y) ∂x ,−y

∂log ¯F(x, y)

∂y )

. (31)

There exists an identity connecting (r1(x, y), r2(x, y)) and (m1(x, y), m2(x, y)). Differentiating

m1(x, y) =x+ 1 ¯

F(x, y)

∞ ∫

x

¯

F(t, y)dt

with respect toxand rearranging terms,

¯

F(x, y)∂m1(x, y)

∂x = (m1(x, y) +x)

∂F¯(x, y)

∂x .

This gives

r1(x, y) =

x∂m1(x,y) ∂x

m1(x, y)−x

(32)

and similarly

r2(x, y) =

y∂m2(x,y) ∂y

m2(x, y)−y

. (33)

A redeeming feature of (r1(x, y), r2(x, y)) is that it allows simple analytically tractable expression for various distributions in the bivariate Pareto family, while the other functions can be expressed in terms of special functions only for many members. See Table 3 for expressions of (r1(x, y), r2(x, y)). It may be noted that the characterizations developed in Section 3 can be transformed in terms of (r1(x, y), r2(x, y)).

From Theorems 4 and 5 and the above deliberations, the following result is ap-parent.

Theorem 13. The following statements are equivalent: a. (X, Y)possessesBDP −1(BDP −2)

b. (i1(x, y), i2(x, y))is constant(locally constant) c. (l1(x, y), l2(x, y))is constant(locally constant) d. (r1(x, y), r2(x, y))is constant(locally constant) e. (X, Y)is distributed asF¯1(x, y)(F¯4(x, y))

(Local constancy of a vector means that the first component is independent of x

acterizations

of

a

family

of

bivariate

Par

eto

distributions

287

TABLE 3

Bivariate generalized failure rates

Distribution

(

r

(

x, y

)

, r

(

x, y

))

¯

F1

(

x, y

)

(

a

=

, b

=

)

¯

F

(

x, y

)

(

axaα xaα_+ybα₋₁

,

bybα xaα_+ybα₋₁

)

¯

F

(

x, y

)

(

ax x+y−1

,

by x+y−1

)

¯

F

(

x, y

)

(

µX(y)

µX(y)−1

,

µY(x)−1

)

¯

F

(

x, y

)

(

σαxα(1+yβ) (xα_+yβ_+xα_yβ₋₁₎

,

σαyβ(1+xα) (xα_+yβ_+xα_yβ₋₁₎

)

¯

F

(

x, y

)

(

ca

(log

x

)

(1

−

(

b

log

y

)

)

, cb

(log

y

)

(1

−

(

a

log

x

)

))

¯

F

(

x, y

)

(1 + 2

x

y

−

x

−

y

)

(

ax

₍₂

_y

−

₁₎

_{, by}

₍₂

_x

−

₁₎

)

¯

F

(

x, y

)

λ

{

λ

a

(log

x

)

+

λ

b

(log

y

)

}

1

α

₍

_a

_(log

_x

₎

_{, b}

_(log

_y

₎

₎

¯

F

(

x, y

)

[

pq

+ (

x

₋

_q

₎₍

_y

₋

_q

₎

]

_ap

(

_x

₍

_y

₋

_q

₎

_{, y}

₍

_x

₋

_q

₎

)

¯

F

(

x, y

)

a

(

x

+

y

+

xy

−

1)

(

x

(1 +

y

)

, y

(1 +

x

))

¯

6. Conclusion

In this paper, we have developed characterizations of a family of bivariate Pareto distributions. The well known dullness property was extended to the bivariate set up and characterizations of bivariate Pareto distributions using this property were derived. The measures of income inequality such as income gap ratio and mean left proportional residual income were proposed and studied in the bivariate case. The generalized failure rate was extended to the bivariate set up and characterizations using this concept were derived. The properties of the family of bivariate Pareto distribtions using copula theory are yet to be studied. The work in this direction will be reported elsewhere.

Acknowledgements

We thank the referee and the editor for the constructive comments. The third author thanks Department of Science and Technology, Government of India for providing financial support.

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Summary

bivariate generalized failure rate useful in reliability analysis. Characterizations, using the above concepts, for various members of the family of bivariate Pareto distributions are derived.